Optimal. Leaf size=527 \[ -\frac {2 \sqrt {a d-b c} (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (4 a^2 C d f^2+a b f (-5 B d f-c C f+3 C d e)-\left (b^2 (5 d f (2 B e-3 A f)-C e (c f+8 d e))\right )\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right ),\frac {f (b c-a d)}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {e+f x}}-\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+3 a C d e-5 A b d f+b c C e)-(2 a d f-b c f+2 b d e) (4 a C d f+b (-5 B d f+2 c C f+4 C d e))) E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (4 a C d f+b (-5 B d f+2 c C f+4 C d e))}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f} \]
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Rubi [A] time = 0.98, antiderivative size = 527, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {1615, 154, 158, 114, 113, 121, 120} \[ -\frac {2 \sqrt {a d-b c} (d e-c f) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} \left (4 a^2 C d f^2+a b f (-5 B d f-c C f+3 C d e)+b^2 (-(5 d f (2 B e-3 A f)-C e (c f+8 d e)))\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {e+f x}}-\frac {2 \sqrt {e+f x} \sqrt {a d-b c} \sqrt {\frac {b (c+d x)}{b c-a d}} (3 b d f (a c C f+3 a C d e-5 A b d f+b c C e)-(2 a d f-b c f+2 b d e) (4 a C d f+b (-5 B d f+2 c C f+4 C d e))) E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {a d-b c}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x} (4 a C d f+b (-5 B d f+2 c C f+4 C d e))}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f} \]
Antiderivative was successfully verified.
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Rule 113
Rule 114
Rule 120
Rule 121
Rule 154
Rule 158
Rule 1615
Rubi steps
\begin {align*} \int \frac {\sqrt {c+d x} \left (A+B x+C x^2\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx &=\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}+\frac {2 \int \frac {\sqrt {c+d x} \left (-\frac {1}{2} b (b c C e+3 a C d e+a c C f-5 A b d f)-\frac {1}{2} b (4 a C d f+b (4 C d e+2 c C f-5 B d f)) x\right )}{\sqrt {a+b x} \sqrt {e+f x}} \, dx}{5 b^2 d f}\\ &=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}+\frac {4 \int \frac {-\frac {1}{4} b (3 b c f (b c C e+3 a C d e+a c C f-5 A b d f)-(b c e+a d e+a c f) (4 a C d f+b (4 C d e+2 c C f-5 B d f)))-\frac {1}{4} b (3 b d f (b c C e+3 a C d e+a c C f-5 A b d f)-(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))) x}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 b^3 d f^2}\\ &=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\left ((d e-c f) \left (4 a^2 C d f^2+a b f (3 C d e-c C f-5 B d f)-b^2 (5 d f (2 B e-3 A f)-C e (8 d e+c f))\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}} \, dx}{15 b^2 d f^3}-\frac {(3 b d f (b c C e+3 a C d e+a c C f-5 A b d f)-(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))) \int \frac {\sqrt {e+f x}}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{15 b^2 d f^3}\\ &=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {\left ((d e-c f) \left (4 a^2 C d f^2+a b f (3 C d e-c C f-5 B d f)-b^2 (5 d f (2 B e-3 A f)-C e (8 d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {e+f x}} \, dx}{15 b^2 d f^3 \sqrt {c+d x}}-\frac {\left ((3 b d f (b c C e+3 a C d e+a c C f-5 A b d f)-(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x}\right ) \int \frac {\sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}}} \, dx}{15 b^2 d f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}\\ &=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {2 \sqrt {-b c+a d} (3 b d f (b c C e+3 a C d e+a c C f-5 A b d f)-(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {\left ((d e-c f) \left (4 a^2 C d f^2+a b f (3 C d e-c C f-5 B d f)-b^2 (5 d f (2 B e-3 A f)-C e (8 d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}}\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {\frac {b c}{b c-a d}+\frac {b d x}{b c-a d}} \sqrt {\frac {b e}{b e-a f}+\frac {b f x}{b e-a f}}} \, dx}{15 b^2 d f^3 \sqrt {c+d x} \sqrt {e+f x}}\\ &=-\frac {2 (4 a C d f+b (4 C d e+2 c C f-5 B d f)) \sqrt {a+b x} \sqrt {c+d x} \sqrt {e+f x}}{15 b^2 d f^2}+\frac {2 C \sqrt {a+b x} (c+d x)^{3/2} \sqrt {e+f x}}{5 b d f}-\frac {2 \sqrt {-b c+a d} (3 b d f (b c C e+3 a C d e+a c C f-5 A b d f)-(2 b d e-b c f+2 a d f) (4 a C d f+b (4 C d e+2 c C f-5 B d f))) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {e+f x} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {\frac {b (e+f x)}{b e-a f}}}-\frac {2 \sqrt {-b c+a d} (d e-c f) \left (4 a^2 C d f^2+a b f (3 C d e-c C f-5 B d f)-b^2 (5 d f (2 B e-3 A f)-C e (8 d e+c f))\right ) \sqrt {\frac {b (c+d x)}{b c-a d}} \sqrt {\frac {b (e+f x)}{b e-a f}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {-b c+a d}}\right )|\frac {(b c-a d) f}{d (b e-a f)}\right )}{15 b^3 d^{3/2} f^3 \sqrt {c+d x} \sqrt {e+f x}}\\ \end {align*}
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Mathematica [C] time = 9.63, size = 562, normalized size = 1.07 \[ \frac {2 \sqrt {a+b x} \left (i b d f \sqrt {a+b x} \sqrt {\frac {b c}{d}-a} (d e-c f) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} (-4 a C d f+5 b B d f-2 b C (c f+2 d e)) \operatorname {EllipticF}\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b c}{d}-a}}{\sqrt {a+b x}}\right ),\frac {b d e-a d f}{b c f-a d f}\right )+\frac {b^2 (c+d x) (e+f x) \left (8 a^2 C d^2 f^2+a b d f (-10 B d f-3 c C f+7 C d e)+b^2 \left (5 d f (3 A d f+B c f-2 B d e)+C \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )\right )\right )}{a+b x}+\frac {i f \sqrt {a+b x} (b c-a d) \sqrt {\frac {b (c+d x)}{d (a+b x)}} \sqrt {\frac {b (e+f x)}{f (a+b x)}} \left (8 a^2 C d^2 f^2+a b d f (-10 B d f-3 c C f+7 C d e)+b^2 \left (5 d f (3 A d f+B c f-2 B d e)+C \left (-2 c^2 f^2-3 c d e f+8 d^2 e^2\right )\right )\right ) E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b c}{d}-a}}{\sqrt {a+b x}}\right )|\frac {b d e-a d f}{b c f-a d f}\right )}{\sqrt {\frac {b c}{d}-a}}+b^2 d f (c+d x) (e+f x) (-4 a C d f+5 b B d f+b C (c f-4 d e+3 d f x))\right )}{15 b^4 d^2 f^3 \sqrt {c+d x} \sqrt {e+f x}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C x^{2} + B x + A\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {f x + e}}{b f x^{2} + a e + {\left (b e + a f\right )} x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{\sqrt {b x + a} \sqrt {f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 6049, normalized size = 11.48 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C x^{2} + B x + A\right )} \sqrt {d x + c}}{\sqrt {b x + a} \sqrt {f x + e}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {c+d\,x}\,\left (C\,x^2+B\,x+A\right )}{\sqrt {e+f\,x}\,\sqrt {a+b\,x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c + d x} \left (A + B x + C x^{2}\right )}{\sqrt {a + b x} \sqrt {e + f x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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